Rational Thermodynamics Research Articles

Relating Entropy Flux With Heat Flux in Two-Temperature Thermodynamic Model for Metal Thermoviscoplasticity

A rigorous derivation of the relation between the entropy flux and the heat flux in a recently developed two-temperature thermodynamic model of metal thermoviscoplasticity is presented. The two-temperature model exploits the internal variable theory of thermodynamics, wherein thermodynamic restrictions on the constitutive functions are based on the second law written in a form similar to the classical Clausius–Duhem (CD) inequality. Here, the weakly interacting thermodynamic subsystems, e.g., configurational and kinetic vibrational subsystems, enable defining their own temperatures, heat fluxes, and entropy fluxes. The CD-type inequality is then constructed with the assumption, as in rational thermodynamics, that entropy fluxes equal heat fluxes divided by respective absolute temperatures. Validity or otherwise of this restrictive assumption is however an open question in the context of two-temperature thermomechanics, and there are, indeed, known materials for which this assumption fails to hold. To settle this important point, we start with a detailed analysis based on a general entropy inequality, whose thermodynamic consequences are extracted using Müller–Liu procedure of Lagrange multipliers, and subsequently, appeal to material frame-indifference, material symmetry groups for additional constitutive restrictions. We conclude that, for isotropic–viscoplastic materials, subsystem entropy fluxes are indeed given by the respective heat fluxes divided by their own temperatures.

Thermomechanical model of nonlocal deformation of a solid

We use relations of rational thermodynamics of irreversible processes for a continuous medium with intrinsic state parameters and Eringen’s model of nonlocal theory of elasticity to study the approach to the construction of mathematical models of thermomechanical processes in a deformable body with regard to the effects of temporal and spatial nonlocality of the continuous medium.

Modeling of Horizontal Water Redistribution in an Unsaturated Soil

Core Ideas A recent laboratory experiment of horizontal water redistribution was simulated. Two different models were separately used to reproduce the measured data. Interfacial area model results showed better agreement with experimental data. When two soil samples with the same hydraulic properties but different initial water saturations are brought into contact, water will redistribute horizontally between the samples until some equilibrium is reached. The part with a higher initial saturation undergoes drainage while imbibition occurs in the other part. Hysteresis will not allow water to redistribute evenly between the two parts. In this study we used two different modeling approaches to analyze a recent experiment related to the water redistribution process. In one approach, we assumed applicability of the standard Richards equation with a hysteretic capillary pressure–saturation relationship (including scanning curves). This approach assumes continuity in the water pressure and flux across the contact surface between the two sides. In the second approach, we used an extended two‐phase flow formulation based on rational thermodynamics principles and involving the air–water specific interfacial area. For this approach, we used continuity in the Gibbs free energy for air–water interfaces and the interfacial area flux as additional conditions at the contact surface. We employed two different initial conditions: uniform initial saturation for each side and slightly nonuniform initial saturation distributions consistent with the measured water contents. We compared results of both models with measurements. The Richards equation with full hysteresis could not reproduce the measured saturation distribution unless an unrealistic value of the imbibition retention curve was assigned. The interfacial area model compared well with the experimental data after optimization of some of the model parameters.

Macroscopic description of capillary transport of liquid and gas in unsaturated porous materials

A new macroscopic description of capillary transport of liquid and gas in porous materials is presented within the framework of multiphase continuum mechanics. It is assumed that unsaturated porous material form a macroscopic continuum composed of three constituents: gas, mobile liquid and capillary liquid, while the skeleton is rigid with an isotropic pore space structure. The capillary liquid forms a thin layer on the internal contact surface with the skeleton, is immoveable and contains the whole capillary energy. The remaining part of the liquid, surrounded by the layer of the capillary liquid and menisci, forms the constituent called mobile liquid. Both liquids exchange mass, momentum and energy in the vicinity of menisci surfaces during their motion, which is described by an additional macroscopic velocity field. A macroscopic scalar quantity is introduced, parametrizing changes of saturations, which for quasi static and stationary processes is equal to the capillary pressure. This quantity extends the dimensionality of the description of mechanical processes taking place in unsaturated porous materials. For the three fluid constituents of the medium, balance equations of mass and linear momentum and evolution equations for saturations are formulated. The constitutive relations for quantities describing mechanical processes in such a medium are proposed based on the dissipation inequality of mechanical energy for the whole system. A new approach is proposed, similar to that used in rational thermodynamics, based on entropy inequality analysis and the Lagrange multipliers method.

Multiscale modeling of lithium ion batteries: thermal aspects.

The thermal behavior of lithium ion batteries has a huge impact on their lifetime and the initiation of degradation processes. The development of hot spots or large local overpotentials leading, e.g., to lithium metal deposition depends on material properties as well as on the nano- und microstructure of the electrodes. In recent years a theoretical structure emerges, which opens the possibility to establish a systematic modeling strategy from atomistic to continuum scale to capture and couple the relevant phenomena on each scale. We outline the building blocks for such a systematic approach and discuss in detail a rigorous approach for the continuum scale based on rational thermodynamics and homogenization theories. Our focus is on the development of a systematic thermodynamically consistent theory for thermal phenomena in batteries at the microstructure scale and at the cell scale. We discuss the importance of carefully defining the continuum fields for being able to compare seemingly different phenomenological theories and for obtaining rules to determine unknown parameters of the theory by experiments or lower-scale theories. The resulting continuum models for the microscopic and the cell scale are numerically solved in full 3D resolution. The complex very localized distributions of heat sources in a microstructure of a battery and the problems of mapping these localized sources on an averaged porous electrode model are discussed by comparing the detailed 3D microstructure-resolved simulations of the heat distribution with the result of the upscaled porous electrode model. It is shown, that not all heat sources that exist on the microstructure scale are represented in the averaged theory due to subtle cancellation effects of interface and bulk heat sources. Nevertheless, we find that in special cases the averaged thermal behavior can be captured very well by porous electrode theory.

Steepest entropy ascent model for far-nonequilibrium thermodynamics: unified implementation of the maximum entropy production principle.

By suitable reformulations, we cast the mathematical frameworks of several well-known different approaches to the description of nonequilibrium dynamics into a unified formulation valid in all these contexts, which extends to such frameworks the concept of steepest entropy ascent (SEA) dynamics introduced by the present author in previous works on quantum thermodynamics. Actually, the present formulation constitutes a generalization also for the quantum thermodynamics framework. The analysis emphasizes that in the SEA modeling principle a key role is played by the geometrical metric with respect to which to measure the length of a trajectory in state space. In the near-thermodynamic-equilibrium limit, the metric tensor is directly related to the Onsager's generalized resistivity tensor. Therefore, through the identification of a suitable metric field which generalizes the Onsager generalized resistance to the arbitrarily far-nonequilibrium domain, most of the existing theories of nonequilibrium thermodynamics can be cast in such a way that the state exhibits the spontaneous tendency to evolve in state space along the path of SEA compatible with the conservation constraints and the boundary conditions. The resulting unified family of SEA dynamical models is intrinsically and strongly consistent with the second law of thermodynamics. The non-negativity of the entropy production is a general and readily proved feature of SEA dynamics. In several of the different approaches to nonequilibrium description we consider here, the SEA concept has not been investigated before. We believe it defines the precise meaning and the domain of general validity of the so-called maximum entropy production principle. Therefore, it is hoped that the present unifying approach may prove useful in providing a fresh basis for effective, thermodynamically consistent, numerical models and theoretical treatments of irreversible conservative relaxation towards equilibrium from far nonequilibrium states. The mathematical frameworks we consider are the following: (A) statistical or information-theoretic models of relaxation; (B) small-scale and rarefied gas dynamics (i.e., kinetic models for the Boltzmann equation); (C) rational extended thermodynamics, macroscopic nonequilibrium thermodynamics, and chemical kinetics; (D) mesoscopic nonequilibrium thermodynamics, continuum mechanics with fluctuations; and (E) quantum statistical mechanics, quantum thermodynamics, mesoscopic nonequilibrium quantum thermodynamics, and intrinsic quantum thermodynamics.

Modelling of thermal diffusion and thermodynamic restrictions

Upon a review of the balance equations for a mixture, diffusion fluxes are shown to satisfy an evolution equation which, with slight differences, describes diffusion relative to a constituent or relative to the barycentric reference. As a consequence, the particular case of binary mixtures is considered with the view of establishing a connection with the current models. The fact that the diffusion flux is governed by a balance (evolution) equation is quite unusual in the literature since customary models are based on constitutive equations (Fick’s law, Soret and Dufour models). This in turn motivates the analysis for a more appropriate thermodynamic setting. The second law for the whole mixture, along with some constitutive assumptions in classical (nonequilibrium) thermodynamics, is shown to provide the constitutive equations which are currently applied in the literature. Instead, the model of diffusion flux through a balance equation proves to be fully consistent with rational thermodynamics and the assumption that the functions, relative to a single constituent, depend on variables pertaining to that constituent whereas interaction terms are allowed to depend also on quantities pertaining to the other constituents.

A note about the rate-and-state-dependent friction model in a thermodynamic framework of the Biot-type equation

The conventional, phenomenological rate-and-state-dependent friction model of Dieterich–Ruina's type is discussed and slightly modified so that, after introducing an artificial internal variable (formally in a position like effective temperature) on the fault, it is driven by a free and a dissipative energies. In contrast to the original model, it thus allows for a formulation in the framework of rational thermodynamics, including the energy balance, and for rigorous numerical analysis. This also suggests an analogous rate-and-state-dependent plastic bulk model using damage/temperature as the state variable controlling the plastic yield stress.

A Thermodynamically Consistent Model for Electric Double Layers in Li All-Solid-State Batteries

Due to their expected improved safety properties, Li all-solid-state batteries provide an attractive alternative to conventional battery systems. However, as yet their performance is strongly limited, since the relevant transport restrictions in the solid electrolyte are still not sufficiently understood. Of great importance in this context are space charge regions at the boundaries between active particles and the electrolyte, similar to the electric double layers in liquid electrolytes. Gaining knowledge about the behavior of this region and the factors that influence it will be a significant step towards a better understanding of the transport properties of a solid electrolyte. In this work, we propose a new, mathematically rigorous model for a Li solid electrolyte. Whereas literature on the modeling of liquid electrolytes is abundantly available, the situation is very different for solid electrolytes. The number of theoretical treatments on this topic is very limited [1], and existing work does not always treat transport and equilibrium properties consistently. We overcome this problem by carefully developing a thermodynamically fully consistent model on the basis of first principles only, using the method of rational thermodynamics. To this end, the laws of continuum mechanics are supplemented by constitutive equations, which can be derived from a free-energy functional that describes the properties of the system on the macroscopic scale (i.e., beyond ~10 interatomic distances). An important step is to identify the proper free-energy functional for the solid electrolyte. Basing our theory on a functional that describes Li ion hopping on a fixed anion lattice containing defects, we analyze the charge and potential distributions in a solid electrolyte in contact with a metal electrode under the influence of varying electrode potential differences. The resulting differential equations are solved numerically as well as analytically in the limit of small electrode potential differences.

Entropy Principle and Recent Results in Non-Equilibrium Theories

We present the state of the art on the modern mathematical methods of exploiting the entropy principle in thermomechanics of continuous media. A survey of recent results and conceptual discussions of this topic in some well-known non-equilibrium theories (Classical irreversible thermodynamics CIT, Rational thermodynamics RT, Thermodynamics of irreversible processes TIP, Extended irreversible thermodynamics EIT, Rational Extended thermodynamics RET) is also summarized.

Consistent theory of mixtures on different levels of description

We provide an extensively discussed formulation of theory of mixtures of interacting fluids. This careful discussion leads to clarification of questions whether kinetic energy of diffusion should be defined to be part of internal energy or not, whether potential energy should be included into internal energy or not, how studying time-reversal parity of quantities determines a correct identification of entropy flux and production, or what definition of partial pressure should be adopted for non-ideal mixtures. Moreover, a generalized chemical affinity is revealed which provides natural means of coupling diffusion with chemical reactions. Finally, the theory is regarded from multilevel point of view providing a means of comparing with theories of mixtures developed within classical irreversible thermodynamics, rational (extended) thermodynamics, extended irreversible thermodynamics or GENERIC.

Numerical solution of integrodifferential heat conduction equation for a nonlocal medium

A heat conduction model is proposed based on the relations of rational thermodynamics of irreversible processes, taking into account the nonlocal nature of the medium and the finite speed of heat propagation. In the one-dimensional case, the numerical solution of the integral-differential equation of heat conduction is obtained by the finite-element method.

Modeling interfacial dynamics using nonequilibrium thermodynamics frameworks

In recent years several nonequilibrium thermodynamic frameworks have been developed capable of describing the dynamics of multiphase systems with complex microstructured interfaces. In this paper we present an overview of these frameworks. We will discuss interfacial dynamics in the context of the classical irreversible thermodynamics, extended irreversible thermodynamics, extended rational thermodynamics, and GENERIC framework, and compare the advantages and disadvantages of these frameworks.

Response to comment on “Averaging theory for description of environmental problems: What have we learned?”

abstract 2012 Elsevier Ltd. All rights reserved. We have identified three distinct elements in the comment byBaveye [2] on our manuscript [22]. These are editorializing, techni-cal critique of content of the paper, and attribution of personalcharacteristics and motivation. We will discuss each of theseaspects of the comment in this response.1. Editorial commentsFirst, the author of [2] editorializes on his perspectives abouttheoretical approaches to porous medium studies. He commentson literatureand aspects of research thatarewelloutside the scopeoftopicswediscussedinourmanuscript(e.g.,statisticalmechanics,Richards’ equation, molecular dynamics). He offers his view onwhat should be the contents of scientific papers. We support theideaofindividualsprovidingthoughtfulcontributionsondirectionsresearch might take or how some particular and specific interestsmight impact broader problems (e.g., [35]) and have made ourown contributions along these lines (e.g., [10,11,28–30]). However,we do not think that such contributions should be couched in whatis purportedly a comment on an article when the topics mentionedin the comment are outside the scope of the article, even totallyunrelated to the article. Those contributions are appropriately pre-pared as stand alone presentations, editorials, or refereed papers.We believewe have no responsibility to be responsive to assertionsin the comment that are unrelated to the scope and thrust of ourpaper. Note that we stated in the article: ‘‘The purpose of this contri-bution is to serve as a basis from which future understanding of aver-aging theory and its applications can develop.’’ This purpose mostcertainlydoesnotdenythefactthatotherapproaches—experimen-tal, theoretical, and computational—are essential to gaining under-standinginanyarea ofintellectualinquiry;ourarticle hasalimitedscope. Although the commenter may be looking for a contributionwith a different purpose or scope from ours, we think that a com-ment on our contribution should confine itself to what we haveactually contributed. We offer no reaction to the editorial proposi-tions comprising most of the comment that are unrelated to ourmanuscript.2. Technical commentsA second element of the comment is concerned with certaintechnical aspects of our contribution. We will identify thoseremarks and respond to them.The comment characterizes our article as ‘‘dealing specificallywith the so-called Thermodynamically Constrained Averaging Theory(TCAT), and ...what has been learned as a result of its develop-ment.’’ Our article actually takes a broader view. Although the TCATapproach is emphasized, as the most recent of averaging methods,the features of the method of volume averaging (MVA) and of aver-aging of conservation equations with rational thermodynamics(ACRT) are also reviewed in brief. A point of emphasis is that ACRTmakes use of symbols for intensive properties that are ill-defined,especially in terms of smaller scale properties. This shortcoming

Lazy zone bone remodeling theory and its relation to topology optimization

The connection between apparent density-type bone remodeling theories and density formulations of topology optimization is well known from a large number of publications and its theoretical basis has recently been discussed by making use of a dynamical systems approach to optimization. The present paper takes this connection one step further by showing how the Coleman–Noll procedure of rational thermodynamics can be used to derive general dynamical systems, where a special case includes the lazy zone concept of bone remodeling theory. It is also shown how a numerical solution method for the dynamical system can be developed by using the sequential convex approximation idea. The method is employed to obtain a series of solutions that show the influence of modeling parameters representing elements of plasticity and viscosity in the growth process.

Mathematical models of thermomechanics of a relaxing solid

Mathematical models of thermomechanic processes based on the laws of rational thermodynamics of irreversible processes are considered. Specific characteristics of the continuum nonstationary behavior are shown in the framework of variousmodels of a medium with internal state parameters.

Partial Pressures in Thermodynamics of Classical Fluid Mixtures

Partial Pressures in Thermodynamics of Classical Fluid Mixtures In the rational thermodynamics of most usual nonequilibrium "classical" fluid mixtures it has been proposed the "thermodynamic" partial pressure which generalize traditional definitions and merge together in an ideal gas mixture. In this paper, these thermodynamic partial pressures are calculated for a (real) gas mixture of methane-ethane-carbon dioxide and a liquid mixture of lithium hydroxide in water. The results are compared with those obtained using the classical formulations of partial pressures calculated in these mixtures as well.

Dynamic properties of interfaces in soft matter: Experiments and theory

The dynamic properties of interfaces often play a crucial role in the macroscopic dynamics of multiphase soft condensed matter systems. These properties affect the dynamics of emulsions, of dispersions of vesicles, of biological fluids, of coatings, of free surface flows, of immiscible polymer blends, and of many other complex systems. The study of interfacial dynamic properties, surface rheology, is therefore a relevant discipline for many branches of physics, chemistry, engineering, and life sciences. In the past three to four decades a vast amount of literature has been produced dealing with the rheological properties of interfaces stabilized by low molecular weight surfactants, proteins, (bio)polymers, lipids, colloidal particles, and various mixtures of these surface active components. In this paper recent experiments are reviewed in the field of surface rheology, with particular emphasis on the models used to analyze surface rheological data. Most of the models currently used are straightforward generalizations of models developed for the analysis of rheological data of bulk phases. In general the limits on the validity of these generalizations are not discussed. Not much use is being made of recent advances in nonequilibrium thermodynamic formalisms for multiphase systems, to construct admissible models for the stress-deformation behavior of interfaces. These formalisms are ideally suited to construct thermodynamically admissible constitutive equations for rheological behavior that include the often relevant couplings to other fluxes in the interface (heat and mass), and couplings to the transfer of mass from the bulk phase to the interface. In this review recent advances in the application of classical irreversible thermodynamics, extended irreversible thermodynamics, rational thermodynamics, extended rational thermodynamics, and the general equation for the nonequilibrium reversible-irreversible coupling formalism to multiphase systems are also discussed, and shown how these formalisms can be used to generate a wide range of thermodynamically admissible constitutive models for the surface stress tensor. Some of the generalizations currently in use are shown to have only limited validity. The aim of this review is to stimulate new developments in the fields of experimental surface rheology and constitutive modeling of multiphase systems using nonequilibrium thermodynamic formalisms and to promote a closer integration of these disciplines.

An extended rational thermodynamics model for surface excess fluxes

In this paper, we derive constitutive equations for the surface excess fluxes in multiphase systems, in the context of an extended rational thermodynamics formalism. This formalism allows us to derive Maxwell–Cattaneo type constitutive laws for the surface extra stress tensor, the surface thermal energy flux vector, and the surface mass flux vector, which incorporate a direct coupling to their corresponding bulk fluxes in the adjacent bulk phases. These constitutive laws also incorporate contributions to the time evolution of the surface excess fluxes from spatial inhomogeneities in these flux fields. These phenomenological equations can be used to model the dynamic behavior of complex viscoelastic interfaces in multiphase systems, in the small deformation limit.

Refined resultant thermomechanics of shells

The resultant two-dimensional (2D) balance laws of mass, linear and angular momentum, and energy as well as the entropy inequality for shells are derived by direct through-the-thickness integration of corresponding 3D laws of continuum thermomechanics. It is indicated that the resultant shell stress power cannot be expressed exactly through the 2D shell stress and strain measures alone. Hence, an additional stress power called an interstitial working is added to the resultant 2D balance of energy. The new, refined, resultant balance of energy and entropy inequality derived here are regarded to be exact implications of corresponding global 3D laws of rational thermodynamics. The kinematic structure of our shell theory is that of the Cosserat surface, while our refined resultant laws of thermomechanics contain three additional surface fields somewhat similar to those present in 3D extended thermodynamics. We briefly analyse the restrictions imposed by our refined resultant entropy inequality on the forms of 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. It is shown, in particular, that in such shells the refined resultant entropy inequality allows one to account for some longer-range spatial interactions. We also present several novel forms of 2D kinetic constitutive equations compatible with the resultant shell equations.